3: Critical Properties and Acentric Factor
Distillation Science (a blend of Chemistry and Chemical Engineering)
This is Part III, Critical Properties and Acentric Factor of a ten-part series of technical articles on Distillation Science, as is currently practiced on an industrial level. See also Part I, Overview for introductory comments, the scope of the article series, and nomenclature. The goal of this article is to explain how the critical properties and acentric factor are used, and how they may be estimated when reliable values are not available.
As discussed in Part II , the more basic vapor pressure equations have limitations that prevent them from being used in many modern industrial distillation systems, specifically at higher pressures. The concepts in this article will be used in Part IV , to introduce an improved vapor pressure equation, which removes the limitations of basic vapor pressure equations.
To implement the more sophisticated VP relationships and inter-compare the constants in Part IV, experimentally measured P vs T data needs to be put into reduced form. This is done only with the knowledge of the critical point, which is that singularity on the saturation line where the liquid and vapor phases become one. The critical temperature and pressure are identified as T c and P c . At the critical point, the specific molar volume is V c .
\[T_{r} = T/T_{c} \label{3-1} \]
\(P_{r} = P/P_{c} \nonumber\)
\(V_{r} = V/V_{c} \nonumber\)
(note that reduced properties have the advantage of being dimensionless)
Using reduced variables T r , P r and V r from Equation ( \ref{3-1} ) also allows introduction to the concept of the "Law of Corresponding States" ( which is really not a scientific law per se, but more of a generally followed relationship). This "law" expresses the generalization that those properties dependent on intermolecular forces are related to the critical properties in the same way for all fluids. This concept underlies the development of several later articles of this series on Distillation Science, including Parts IV through VII .
The critical compressibility is Z c , as defined as:
\[Z_{c}= \frac {P_{c}V_{c}}{RT_{c}} \label{3-2} \] which is also dimensionless.
Two other properties are important to note:
- \(T_b\) is defined as the atmospheric pressure boiling point of a fluid, with \(T_{br}\) as the reduced value of the atmospheric boiling point
- \(ω\) is the acentric factor, which is a measure of molecular complexity as defined as:
\[\omega=-\log_{10}(VP)-1 \label{3-3} \] where the VP (in atmospheres) is evaluated at T r = 0.7
The term “acentric factor” comes from Kenneth Pfizer's observation in 1955 that compact and near-spherical molecules have close to zero values when their ω is calculated. For example, neon, argon and krypton have ω values of -0.04, 0.00 and 0.00, respectively (helium has a more slightly negative ω, the value depending on isotope). Methane (CH 4 ) has a ω of 0.011; and the molecularly larger silane (SiH 4 ) has an ω of 0.099. An even larger and more complex molecule, like carbon tetrachloride (CCl 4 ) has an ω of 0.193; whereas silicon tetrachloride (SiCl 4 ) has an ω of 0.248. So valuation of the acentric factor from VP data and knowledge of the critical point is a good validation check against the known molecular size and shape.
For Equations of State ( see Part V ) that are more advanced than Van der Waals, the acentric factor is a required property, along with critical temperature and pressure. It is also used along with critical properties in the estimation of binary interaction parameters in Part VII .
As mentioned in Part II , the basic VP relationships have acceptable accuracy to obtain good values of T b from available data. Experimental data is often taken near atmospheric pressure, but not exactly at one atmosphere absolute (i.e, 760 mmHg = 760 Torr = 101.325 Pa). Instead of blindly accepting a value of T b from a handbook or single website, the practicing scientist or engineer should consider the data’s source and reliability. There is ready internet availability of atmospheric boiling points, critical property data, and acentric factors, especially on the NIST WebBook and global websites like Dechema , Infotherm (recently acquired by John Wiley and Sons from FIZ Chemie Berlin).
A recommended way to sort through the dizzying amount of critical property data (i.e., T c , P c , and V c ) is to use the correlation techniques of Lyderson ( "Estimation of Critical Properties of Organic Compounds", University of Wisconsin College of Engineering, 1955) and others, which are specifically geared toward organic compounds. With some algebraic manipulation and adjustment of Lyderson’s parameters (since the usage intent here is for polar compounds that are not organic, but are not ionic either), and based on data across homologues, the following relationships seem to hold and are recommended for both validating questionable data or filling in “holes” of no data:
For T b , the general correlating relationship within a homologue ( e.g., from silane to silicon tetrachloride, or from phosphine to trichlorophosphine) :
\[(T_{b}\times MW)^n= A+B\times MW \label{3-4} \]
where MW is the molecular weight. For chlorosilanes, chloromethanes, chlorophosphines, etc, this relationship shows a best fit with an exponent “n” of 0.8320. The constants A and B for the homologue are typically set by the hydride and chloride, since that data is usually more reliable. Where the core atom is not just carbon or silicon, but a blend (i.e, a methyl silane), an exponent of 0.8455 fits the data a bit better. This contrasts with organic compounds, whose data tends to fit best with somewhat lower exponents that are closer to 0.80.
For T c (critical temperature), the best correlation within a homologue is found to be:
\[\frac {T_{c}-T_{b}}{T_{c}}= A+B\times MW \label{3-5} \]
where MW is the molecular weight and the constants A and B for the homologue are typically set by the hydride and chloride, as long as there is no organic content and the fluids are monomeric.
For P c (critical temperature), use
\[(\frac {MW}{P_{c}})^n= A+B\times MW \label{3-6} \]
where MW is the molecular weight, exponent "n" = 0.5672, and constants A and B for the homologue are typically set by the hydride and chloride (as long as there is no organic content and the fluids are monomeric). For a pure organic compound, the exponent "n" should be Lyderson’s suggested 0.5000; for methyl silanes (e.g, dimethyl silane) the best exponent is 0.4550; and for Group III (dimeric bridge-bonded) compounds the best exponent fit is 1.03-1.08 ( 1.03 for di-gallanes and 1.08 for diboranes).
For V c (critical molar volume), Lyderson’s rule of
\[V_{c}=A+B\times MW \label{3-7} \]
is probably still the best, where MW is the molecular weight and constants A and B for the homologue are typically set by the hydride and chloride. For some homologues, there is a possibility that the Vc vs MW relationship is not exactly linear, but rather has a slight concave quadratic nature; but the data is rarely good enough to determine such. Of all the critical properties, V c is the hardest to experimentally measure, and is frequently estimated.
There is a way to get around some of the uncertainty with V c , and that is to validate (or make minor V c adjustments) based on the pattern of Z c values in the homologue where
\[Z_{c}= \frac{P_{c}\times V_{c}}{R\times T_{c}} \label{3-8} \]
Within each homologue there is a characteristic “check-mark” pattern to Z c values, when plotted against the number of hydrogen to chlorine swaps of the homologue (i.e., the homologue’s hydride has zero swaps, the monochloride one swap, the dichloride two swaps, etc). See below example data plot Figure 3-1 for the chlorosilane and chlorophosphine homologues.
The homologue’s hydride (which tends to be a fairly spherical shaped molecule with minimal dipole moment) typically has a Z c in the range of 0.27-0.29. The first swap replaces a small “H” atom with a larger “Cl” atom, and re-orients the molecular shape to have a significant dipole, dropping the Z c value by about 10%. Then the next swaps reduce the dipole, until the molecular shape returns closer to spherical, although significantly larger in diameter. Typically the Z c of the completely chlorinated homologue fluid has a slightly larger Z c than the homologue’s hydride.
To the extent that this “checkmark” shape is not observed in plotting calculated Z c values per Equation 3-8, the “culprit” is almost always the value of V c , which allows for some correction. The value of Z c is known to have a significant relationship to molecular shape and complexity, although the stronger relationship between Z c and the acentric factor, ω, is not so easily generalized as in organic compounds.
The importance of accurate values of T b , T c , P c (and ω to an extent) is in calculating vapor pressure using the advanced relationship of Part IV . Having an accurate value of ω is more important, along with T c , P c in calculating the Equation of State for a fluid, in Part V . Having good values for ω and V c are important to the estimation of binary interaction parameters in Part VII .
When two different types of swaps are made in a slightly polar molecule, an alternate technique is used to best correlate fluid properties (T b , T c , P c , V c , and ω), with a good example being methyl chlorosilanes. With methyl chlorosilanes, one or more methyl groups are substituted for hydrogen atoms (i.e., Si-H swapped to Si-CH 3 ), with the remaining Si-H bonding possibly swapped to Si-Cl. Another example would be swapping C-H for C-Cl and C-F bonding in chloro-fluoro methanes. There is no concise formula that governs such a dual substitution. Instead the concept of neural networks is used.
In this technique, the two base curves are laid out from the same starting compound (silane in the example below), and on each base curve the pertinent physical property values shown for each substituted homologue (chlorosilanes and methyl silanes being the base curves in the example below) vs molecular weight. For each amount of combined substitution (100%, 75% and 50% substitution = 0%, 25%, and 50% Si-H remaining) the data points are connected to their respective base curves. For example, on the 75% substitution amount, one would curve-connect trichlorosilane, methyl dichlorosilane, dimethyl monochlorosilane, and trimethyl silane.
Additional curve-connections could be made for values of constant amount of chlorine or methyl substitution (e.g., curve-connecting monochlorosilane, methyl monochlorosilane, dimethyl monochlorosilane, and trimethyl monochlorosilane). Using this method, the data on the dual-substituted fluids can be validated and adjustments made so that the curve-connections are smooth.
In the example in below Figure 3-2, the values of P c for methyl monochlorosilane ( sic , me-mono) and dimethyl monochlorosilane ( sic , di-me mono) are hardened up from rough measurements.
The value of the neural network technique is to make adjustments where data is poor or the estimation is of questionable accuracy. The presumption of neural networks is that for a class of compound, the continuum of structural changes should be uniform and internally consistent (although not necessarily linear).
Neural networks were constructed for all the pertinent properties of methyl chlorosilanes, and the technique found to be rather simple to deal with graphically using MS Excel. The resulting values were included in Table 3-3 below, for chlorosilanes and their common impurities, whose properties were validated using the correlating techniques of Equation ( \ref{3-4} ) through Equation ( \ref{3-8} ) .
|
Fluid |
MW |
T b |
T c |
P c |
V c |
Z c |
ω |
|---|---|---|---|---|---|---|---|
|
Group III(A) |
|||||||
|
B 2 H 6 |
27.670 |
180.54 |
289.70 |
39.58 |
173.10 |
0.2882 |
0.1254 |
|
B 2 H 5 Cl |
62.119 ǂ |
216.37 |
346.33 |
37.13 |
189.92 |
0.2481 |
0.1283 |
|
BH 2 Cl as monomer |
42.284 |
237.72 |
379.56 |
37.10 |
206.73 |
0.2463 |
0.1315 |
|
BHCl 2 as monomer |
82.722 |
266.07 |
422.71 |
37.63 |
240.37 |
0.2608 |
0.1389 |
|
B 2 HCl 5 hypothetic |
199.893 ǂ |
276.67 |
438.47 |
37.92 |
257.18 |
0.2711 |
0.1433 |
|
BCl 3 |
117.169 |
285.88 |
451.95 |
38.20 |
274.00 |
0.2822 |
0.1468 |
|
AlCl 3 as monomer |
133.341 |
466.86 * |
625.70 |
26.00 |
261.80 |
0.1326 |
0.3474 |
|
Ga 2 H 4 Cl 2 as dimer |
214.383 ǂ |
350.86 |
545.93 |
40.94 |
231.9 |
0.2120 |
0.2726 |
|
Ga 2 H 2 Cl 4 as dimer |
283.273 ǂ |
421.79 |
639.69 |
41.02 |
247.5 |
0.1943 |
0.3525 |
|
Ga 2 Cl 6 as dimer |
352.162 ǂ |
473.49 |
694.00 |
37.70 |
263.0 |
0.1741 |
0.4504 |
|
Group IV(A) |
|||||||
|
SiH 4 |
32.117 |
161.75 |
269.65 |
47.99 |
130.07 |
0.2821 |
0.09860 |
|
SiH 3 Cl |
66.562 |
242.75 |
396.65 |
47.82 |
169.32 |
0.2488 |
0.1252 |
|
SiH 2 Cl 2 |
101.007 |
281.45 |
449.45 |
44.83 |
215.05 |
0.2614 |
0.1589 |
|
SiHCl 3 |
135.452 |
306.15 |
479.15 |
41.15 |
267.28 |
0.2797 |
0.2090 |
|
SiCl 4 |
169.896 |
330.72 |
506.95 |
36.50 |
326.00 |
0.2860 |
0.2482 |
|
SiH 3 (CH 3 ) |
46.144 |
216.48 |
348.35 |
41.53 |
185.20 |
0.2691 |
0.1264 |
|
SiH 2 Cl(CH 3 ) |
80.589 |
278.82 |
439.11 |
41.16 |
233.90 |
0.2672 |
0.1793 |
|
SiHCl 2 (CH 3 ) |
115.034 |
314.17 |
489.18 |
39.25 |
287.81 |
0.2814 |
0.2269 |
|
SiCl 3 (CH 3 ) |
149.479 |
339.72 |
517.61 |
35.86 |
345.70 |
0.2919 |
0.2655 |
|
SiH 2 (CH 3 ) 2 |
60.169 |
252.86 |
399.17 |
36.05 |
245.84 |
0.2706 |
0.1604 |
|
SiHCl(CH 3 ) 2 |
94.615 |
305.30 |
471.35 |
35.94 |
300.32 |
0.2791 |
0.2264 |
|
SiCl 2 (CH 3 ) 2 |
129.061 |
342.89 |
519.21 |
34.26 |
358.20 |
0.2880 |
0.2740 |
|
GeH 4 |
76.642 |
184.93 |
307.98 |
54.77 |
128.28 |
0.2780 |
0.1270 |
|
GeH 3 Cl |
111.087 |
302.91 |
495.10 |
49.95 |
180.17 |
0.2215 |
0.1476 |
|
GeH 2 Cl 2 |
145.532 |
334.60 |
536.93 |
45.36 |
236.71 |
0.2437 |
0.1721 |
|
GeHCl 3 |
179.976 |
343.54 |
541.41 |
41.42 |
283.96 |
0.2647 |
0.2012 |
|
GeCl 4 |
214.421 |
357.28 |
553.16 |
38.10 |
335.86 |
0.2819 |
0.2334 |
|
SnH 4 |
122.742 |
221.07 |
360.20 |
51.70 |
152.90 |
0.2674 |
0.1619 |
|
SnCl 4 |
260.521 |
387.21 |
591.85 |
36.95 |
351.20 |
0.2672 |
0.2625 |
|
Group V(A) |
|||||||
|
PH 3 |
33.998 |
185.41 |
324.75 |
64.51 |
113.33 |
0.2743 |
0.03052 |
|
PH 2 Cl |
68.443 |
273.09 |
463.73 |
62.29 |
153.63 |
0.2515 |
0.0750 |
|
PHCl 2 |
102.888 |
318.44 |
524.74 |
55.82 |
202.52 |
0.2625 |
0.1362 |
|
PCl 3 |
137.333 |
349.25 |
558.95 |
50.00 |
260.00 |
0.2834 |
0.2117 |
|
POCl 3 |
153.331 |
379.00 |
605.21 |
47.59 |
276.00 |
0.2645 |
0.1993 |
Table 3-3, Continued
|
AsH 3 |
77.945 |
210.73 |
373.00 |
65.12 |
132.50 |
0.2819 |
0.01341 |
|
AsH 2 Cl |
112.390 |
300.41 |
515.99 |
64.33 |
174.85 |
0.2657 |
0.0573 |
|
AsHCl 2 |
146.835 |
359.66 |
600.00 |
61.57 |
218.29 |
0.2730 |
0.1153 |
|
AsCl 3 |
181.281 |
403.30 |
654.00 |
58.35 |
259.56 |
0.2822 |
0.1875 |
|
SbH 3 |
124.781 |
256.09 |
446.20 |
66.61 |
157.20 |
0.2860 |
0.01659 |
|
SbCl 3 |
228.115 |
794.05 |
794.05 |
68.85 |
268.00 |
0.2832 |
0.3309 |
|
n-pentane |
72.149 |
309.16 |
470.05 |
32.86 |
310.0 |
0.2641 |
0.2393 |
|
iso-pentane |
72.149 |
300.82 |
460.56 |
33.17 |
307.1 |
.2695 |
0.2147 |
In above Table 3-3, those entries that are marked with a ǂ have their molecular weight given as the dimer. For AlCl 3 , the asterisk on the T b entry is for the best correlating value for the Part IV VP equation, even though it is below the triple point. Group IIIA compounds can feature “bridge-bonding” and can be either monomeric or dimeric. The two pentane entries at the end of the table are included because of their significance as electronic impurities, and to show that the techniques can be extended to organics.
With Group IIIA, some compounds are not included in a homologue simply because they are impossible structurally or are unstable. An example would be B 2 H 3 Cl 3 , which would put too much strain on the B-B bridge-bond. Another is AlH 3 which cannot exist as a stand-alone liquid with vapor pressure at normal processing conditions, but tends to form stable hydrides with Group I metals such as LiAlH 4 . However, B 2 HCl 5 is included (but noted as hypothetic), since some researchers claim to have seen its presence in low levels of TCS.


2.1: Development of the Periodic Table
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Learning Objectives
- To become familiar with the history of the periodic table.
The modern periodic table has evolved through a long history of attempts by chemists to arrange the elements according to their properties as an aid in predicting chemical behavior. One of the first to suggest such an arrangement was the German chemist Johannes Dobereiner (1780–1849), who noticed that many of the known elements could be grouped in triads (a set of three elements that have similar properties) —for example, chlorine, bromine, and iodine; or copper, silver, and gold. Dobereiner proposed that all elements could be grouped in such triads, but subsequent attempts to expand his concept were unsuccessful. We now know that portions of the periodic table—the d block in particular—contain triads of elements with substantial similarities. The middle three members of most of the other columns, such as sulfur, selenium, and tellurium in group 16 or aluminum, gallium, and indium in Group 13, also have remarkably similar chemistry.
By the mid-19th century, the atomic masses of many of the elements had been determined. The English chemist John Newlands (1838–1898), hypothesizing that the chemistry of the elements might be related to their masses, arranged the known elements in order of increasing atomic mass and discovered that every seventh element had similar properties (Figure \(\PageIndex{1}\)). Newlands therefore suggested that the elements could be classified into octaves . He described octaves as a group of seven elements which correspond to the horizontal rows in the main groups of today's periodic table. There were seven elements because the noble gases were not known at the time. Unfortunately, Newlands’s “law of octaves” did not seem to work for elements heavier than calcium, and his idea was publicly ridiculed. At one scientific meeting, Newlands was asked why he didn’t arrange the elements in alphabetical order instead of by atomic mass, since that would make just as much sense! Actually, Newlands was on the right track—with only a few exceptions, atomic mass does increase with atomic number, and similar properties occur every time a set of ns 2 np 6 subshells is filled. Despite the fact that Newlands’s table had no logical place for the d -block elements, he was honored for his idea by the Royal Society of London in 1887.
John Newlands (1838–1898)
John Alexander Reina Newlands was an English chemist who worked on the development of the periodic table. He noticed that elemental properties repeated every seventh (or multiple of seven) element, as musical notes repeat every eighth note.
The periodic table achieved its modern form through the work of the German chemist Julius Lothar Meyer (1830–1895) and the Russian chemist Dimitri Mendeleev (1834–1907), both of whom focused on the relationships between atomic mass and various physical and chemical properties. In 1869, they independently proposed essentially identical arrangements of the elements. Meyer aligned the elements in his table according to periodic variations in simple atomic properties, such as “atomic volume” (Figure \(\PageIndex{2}\)), which he obtained by dividing the atomic mass (molar mass) in grams per mole by the density (\(\rho\)) of the element in grams per cubic centimeter. This property is equivalent to what is today defined as molar volume, t he molar mass of an element divided by its density, (measured in cubic centimeters per mole):
\[ \dfrac{molar\; mass\left ( \cancel{g}/mol \right )}{density\left ( \cancel{g}/cm^{3} \right )}=molar\; volume\left ( cm^{3}/mol \right ) \label{7.1.1} \]
As shown in Figure \(\PageIndex{2}\), the alkali metals have the highest molar volumes of the solid elements. In Meyer’s plot of atomic volume versus atomic mass, the nonmetals occur on the rising portion of the graph, and metals occur at the peaks, in the valleys, and on the downslopes.
Dimitri Mendeleev (1834–1907)
When his family’s glass factory was destroyed by fire, Mendeleev moved to St. Petersburg, Russia, to study science. He became ill and was not expected to recover, but he finished his PhD with the help of his professors and fellow students.
In addition to the periodic table, another of Mendeleev’s contributions to science was an outstanding textbook, The Principles of Chemistry , which was used for many years.


6.2: Determining Empirical and Molecular Formulas
- Compute the percent composition of a compound
- Determine the empirical formula of a compound
- Determine the molecular formula of a compound
In the previous section, we discussed the relationship between the bulk mass of a substance and the number of atoms or molecules it contains (moles). Given the chemical formula of the substance, we were able to determine the amount of the substance (moles) from its mass, and vice versa. But what if the chemical formula of a substance is unknown? In this section, we will explore how to apply these very same principles in order to derive the chemical formulas of unknown substances from experimental mass measurements.
Percent Composition
The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most succinct way of representing this elemental makeup. When a compound’s formula is unknown, measuring the mass of each of its constituent elements is often the first step in the process of determining the formula experimentally. The results of these measurements permit the calculation of the compound’s percent composition , defined as the percentage by mass of each element in the compound. For example, consider a gaseous compound composed solely of carbon and hydrogen. The percent composition of this compound could be represented as follows:
\[\mathrm{\%H=\dfrac{mass\: H}{mass\: compound}\times100\%} \nonumber \]
\[\mathrm{\%C=\dfrac{mass\: C}{mass\: compound}\times100\%} \nonumber \]
If analysis of a 10.0-g sample of this gas showed it to contain 2.5 g H and 7.5 g C, the percent composition would be calculated to be 25% H and 75% C:
\[\mathrm{\%H=\dfrac{2.5\:g\: H}{10.0\:g\: compound}\times100\%=25\%} \nonumber \]
\[\mathrm{\%C=\dfrac{7.5\:g\: C}{10.0\:g\: compound}\times100\%=75\%} \nonumber \]
Analysis of a 12.04-g sample of a liquid compound composed of carbon, hydrogen, and nitrogen showed it to contain 7.34 g C, 1.85 g H, and 2.85 g N. What is the percent composition of this compound?
Solution
To calculate percent composition, we divide the experimentally derived mass of each element by the overall mass of the compound, and then convert to a percentage:
\[\mathrm{\%C=\dfrac{7.34\:g\: C}{12.04\:g\: compound}\times100\%=61.0\%} \nonumber \]
\[\mathrm{\%H=\dfrac{1.85\:g\: H}{12.04\:g\: compound}\times100\%=15.4\%} \nonumber \]
\[\mathrm{\%N=\dfrac{2.85\:g\: N}{12.04\:g\: compound}\times100\%=23.7\%} \nonumber \]
The analysis results indicate that the compound is 61.0% C, 15.4% H, and 23.7% N by mass.
A 24.81-g sample of a gaseous compound containing only carbon, oxygen, and chlorine is determined to contain 3.01 g C, 4.00 g O, and 17.81 g Cl. What is this compound’s percent composition?
- Answer
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12.1% C, 16.1% O, 71.8% Cl
Determining Percent Composition from Formula Mass
Percent composition is also useful for evaluating the relative abundance of a given element in different compounds of known formulas. As one example, consider the common nitrogen-containing fertilizers ammonia (NH 3 ), ammonium nitrate (NH 4 NO 3 ), and urea (CH 4 N 2 O). The element nitrogen is the active ingredient for agricultural purposes, so the mass percentage of nitrogen in the compound is a practical and economic concern for consumers choosing among these fertilizers. For these sorts of applications, the percent composition of a compound is easily derived from its formula mass and the atomic masses of its constituent elements. A molecule of NH 3 contains one N atom weighing 14.01 amu and three H atoms weighing a total of (3 × 1.008 amu) = 3.024 amu. The formula mass of ammonia is therefore (14.01 amu + 3.024 amu) = 17.03 amu, and its percent composition is:
\[\mathrm{\%N=\dfrac{14.01\:amu\: N}{17.03\:amu\:NH_3}\times100\%=82.27\%} \nonumber \]
\[\mathrm{\%H=\dfrac{3.024\:amu\: N}{17.03\:amu\:NH_3}\times100\%=17.76\%} \nonumber \]
This same approach may be taken considering a pair of molecules, a dozen molecules, or a mole of molecules, etc. The latter amount is most convenient and would simply involve the use of molar masses instead of atomic and formula masses, as demonstrated Example \(\PageIndex{2}\). As long as we know the chemical formula of the substance in question, we can easily derive percent composition from the formula mass or molar mass.
Aspirin is a compound with the molecular formula C 9 H 8 O 4 . What is its percent composition?
Solution
To calculate the percent composition, we need to know the masses of C, H, and O in a known mass of C 9 H 8 O 4 . It is convenient to consider 1 mol of C 9 H 8 O 4 and use its molar mass (180.159 g/mole, determined from the chemical formula) to calculate the percentages of each of its elements:
\[\begin{align*}
\%\ce C&=\mathrm{\dfrac{9\:mol\: C\times molar\: mass\: C}{molar\: mass\:\ce{C9H18O4}}\times100=\dfrac{9\times12.01\:g/mol} \nonumber{180.159\:g/mol}\times100=\dfrac{108.09\:g/mol}{180.159\:g/mol}\times100} \nonumber\\
\%\ce C&=\mathrm{60.00\,\%\,C} \nonumber
\end{align*} \nonumber \]
\[\begin{align*}
\%\ce H&=\mathrm{\dfrac{8\:mol\: H\times molar\: mass\: H}{molar\: mass\:\ce{C9H18O4}}\times 100=\dfrac{8\times 1.008\:g/mol} \nonumber {180.159\:g/mol}\times 100=\dfrac{8.064\:g/mol}{180.159\:g/mol}\times 100} \nonumber\\
\%\ce H&=4.476\,\%\,\ce H \nonumber
\end{align*} \nonumber \]
\[\begin{align*}
\%\ce O&=\mathrm{\dfrac{4\:mol\: O\times molar\: mass\: O}{molar\: mass\: \ce{C9H18O4}}\times 100=\dfrac{4\times 16.00\:g/mol} \nonumber{180.159\:g/mol}\times 100=\dfrac{64.00\:g/mol}{180.159\:g/mol}\times 100} \nonumber \\
\%\ce O&=35.52\% \nonumber
\end{align*} \nonumber \]
Note that these percentages sum to equal 100.00% when appropriately rounded.
To three significant digits, what is the mass percentage of iron in the compound \(Fe_2O_3\)?
- Answer
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69.9% Fe
Determination of Empirical Formulas
As previously mentioned, the most common approach to determining a compound’s chemical formula is to first measure the masses of its constituent elements. However, we must keep in mind that chemical formulas represent the relative numbers , not masses, of atoms in the substance. Therefore, any experimentally derived data involving mass must be used to derive the corresponding numbers of atoms in the compound. To accomplish this, we can use molar masses to convert the mass of each element to a number of moles. We then consider the moles of each element relative to each other, converting these numbers into a whole-number ratio that can be used to derive the empirical formula of the substance. Consider a sample of compound determined to contain 1.71 g C and 0.287 g H. The corresponding numbers of atoms (in moles) are:
\[\mathrm{1.71\:g\: C\times \dfrac{1\:mol\: C}{12.01\:g\: C}=0.142\:mol\: C} \nonumber \]
\[\mathrm{0.287\:g\: H\times \dfrac{1\:mol\: H}{1.008\:g\: H}=0.284\:mol\: H} \nonumber \]
Thus, we can accurately represent this compound with the formula C 0.142 H 0.284 . Of course, per accepted convention, formulas contain whole-number subscripts, which can be achieved by dividing each subscript by the smaller subscript:
\[\ce C_{\Large{\frac{0.142}{0.142}}}\:\ce H_{\Large{\frac{0.284}{0.142}}}\ce{\:or\:CH2} \nonumber \]
(Recall that subscripts of “1” are not written, but rather assumed if no other number is present.)
The empirical formula for this compound is thus CH 2 . This may or not be the compound’s molecular formula as well; however, we would need additional information to make that determination (as discussed later in this section).
Consider as another example a sample of compound determined to contain 5.31 g Cl and 8.40 g O. Following the same approach yields a tentative empirical formula of:
\[\mathrm{Cl_{0.150}O_{0.525}=Cl_{\Large{\frac{0.150}{0.150}}}\: O_{\Large{\frac{0.525}{0.150}}}=ClO_{3.5}} \nonumber \]
In this case, dividing by the smallest subscript still leaves us with a decimal subscript in the empirical formula. To convert this into a whole number, we must multiply each of the subscripts by two, retaining the same atom ratio and yielding Cl 2 O 7 as the final empirical formula.
In summary, empirical formulas are derived from experimentally measured element masses by:
- Deriving the number of moles of each element from its mass
- Dividing each element’s molar amount by the smallest molar amount to yield subscripts for a tentative empirical formula
- Multiplying all coefficients by an integer, if necessary, to ensure that the smallest whole-number ratio of subscripts is obtained
Figure \(\PageIndex{1}\) outlines this procedure in flow chart fashion for a substance containing elements A and X.
A sample of the black mineral hematite (Figure \(\PageIndex{2}\)), an oxide of iron found in many iron ores, contains 34.97 g of iron and 15.03 g of oxygen. What is the empirical formula of hematite?
Solution
For this problem, we are given the mass in grams of each element. Begin by finding the moles of each:
\[\begin{align*}
\mathrm{34.97\:g\: Fe\left(\dfrac{mol\: Fe}{55.85\:g}\right)}&=\mathrm{0.6261\:mol\: Fe} \nonumber\\ \nonumber\\
\mathrm{15.03\:g\: O\left(\dfrac{mol\: O}{16.00\:g}\right)}&=\mathrm{0.9394\:mol\: O} \nonumber
\end{align*} \nonumber \]
Next, derive the iron-to-oxygen molar ratio by dividing by the lesser number of moles:
\[\mathrm{\dfrac{0.6261}{0.6261}=1.000\:mol\: Fe} \nonumber \]
\[\mathrm{\dfrac{0.9394}{0.6261}=1.500\:mol\: O} \nonumber \]
The ratio is 1.000 mol of iron to 1.500 mol of oxygen (Fe 1 O 1.5 ). Finally, multiply the ratio by two to get the smallest possible whole number subscripts while still maintaining the correct iron-to-oxygen ratio:
\[\mathrm{2(Fe_1O_{1.5})=Fe_2O_3} \nonumber \]
The empirical formula is \(Fe_2O_3\).
What is the empirical formula of a compound if a sample contains 0.130 g of nitrogen and 0.370 g of oxygen?
- Answer
-
\(N_2O_5\)
Video \(\PageIndex{1}\) : Additional worked examples illustrating the derivation of empirical formulas are presented in the brief video clip.
Deriving Empirical Formulas from Percent Composition
Finally, with regard to deriving empirical formulas, consider instances in which a compound’s percent composition is available rather than the absolute masses of the compound’s constituent elements. In such cases, the percent composition can be used to calculate the masses of elements present in any convenient mass of compound; these masses can then be used to derive the empirical formula in the usual fashion.
The bacterial fermentation of grain to produce ethanol forms a gas with a percent composition of 27.29% C and 72.71% O (Figure \(\PageIndex{3}\)). What is the empirical formula for this gas?
Solution
Since the scale for percentages is 100, it is most convenient to calculate the mass of elements present in a sample weighing 100 g. The calculation is “most convenient” because, per the definition for percent composition, the mass of a given element in grams is numerically equivalent to the element’s mass percentage. This numerical equivalence results from the definition of the “percentage” unit, whose name is derived from the Latin phrase per centum meaning “by the hundred.” Considering this definition, the mass percentages provided may be more conveniently expressed as fractions:
\[\begin{align*}
27.29\,\%\,\ce C&=\mathrm{\dfrac{27.29\:g\: C}{100\:g\: compound}} \nonumber \\ \nonumber \\
72.71\,\%\,\ce O&=\mathrm{\dfrac{72.71\:g\: O}{100\:g\: compound}} \nonumber
\end{align*} \nonumber \]
The molar amounts of carbon and hydrogen in a 100-g sample are calculated by dividing each element’s mass by its molar mass:
\[\begin{align*}
\mathrm{27.29\:g\: C\left(\dfrac{mol\: C}{12.01\:g}\right)}&=\mathrm{2.272\:mol\: C} \nonumber \\ \nonumber \\
\mathrm{72.71\:g\: O\left(\dfrac{mol\: O}{16.00\:g}\right)}&=\mathrm{4.544\:mol\: O} \nonumber
\end{align*} \nonumber \]
Coefficients for the tentative empirical formula are derived by dividing each molar amount by the lesser of the two:
\[\mathrm{\dfrac{2.272\:mol\: C}{2.272}=1} \nonumber \]
\[\mathrm{\dfrac{4.544\:mol\: O}{2.272}=2} \nonumber \]
Since the resulting ratio is one carbon to two oxygen atoms, the empirical formula is CO 2 .
What is the empirical formula of a compound containing 40.0% C, 6.71% H, and 53.28% O?
- Answer
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\(CH_2O\)
Derivation of Molecular Formulas
Recall that empirical formulas are symbols representing the relative numbers of a compound’s elements. Determining the absolute numbers of atoms that compose a single molecule of a covalent compound requires knowledge of both its empirical formula and its molecular mass or molar mass. These quantities may be determined experimentally by various measurement techniques. Molecular mass, for example, is often derived from the mass spectrum of the compound (see discussion of this technique in the previous chapter on atoms and molecules). Molar mass can be measured by a number of experimental methods, many of which will be introduced in later chapters of this text.
Molecular formulas are derived by comparing the compound’s molecular or molar mass to its empirical formula mass . As the name suggests, an empirical formula mass is the sum of the average atomic masses of all the atoms represented in an empirical formula. If we know the molecular (or molar) mass of the substance, we can divide this by the empirical formula mass in order to identify the number of empirical formula units per molecule, which we designate as n :
\[\mathrm{\dfrac{molecular\: or\: molar\: mass\left(amu\: or\:\dfrac{g}{mol}\right)}{empirical\: formula\: mass\left(amu\: or\:\dfrac{g}{mol}\right)}= \mathit n\: formula\: units/molecule} \nonumber \]
The molecular formula is then obtained by multiplying each subscript in the empirical formula by n , as shown by the generic empirical formula A x B y :
\[\mathrm{(A_xB_y)_n=A_{nx}B_{nx}} \nonumber \]
For example, consider a covalent compound whose empirical formula is determined to be CH 2 O. The empirical formula mass for this compound is approximately 30 amu (the sum of 12 amu for one C atom, 2 amu for two H atoms, and 16 amu for one O atom). If the compound’s molecular mass is determined to be 180 amu, this indicates that molecules of this compound contain six times the number of atoms represented in the empirical formula:
\[\mathrm{\dfrac{180\:amu/molecule}{30\:\dfrac{amu}{formula\: unit}}=6\:formula\: units/molecule} \nonumber \]
Molecules of this compound are then represented by molecular formulas whose subscripts are six times greater than those in the empirical formula:
\[\ce{(CH2O)6}=\ce{C6H12O6} \nonumber \]
Note that this same approach may be used when the molar mass (g/mol) instead of the molecular mass (amu) is used. In this case, we are merely considering one mole of empirical formula units and molecules, as opposed to single units and molecules.
Nicotine, an alkaloid in the nightshade family of plants that is mainly responsible for the addictive nature of cigarettes, contains 74.02% C, 8.710% H, and 17.27% N. If 40.57 g of nicotine contains 0.2500 mol nicotine, what is the molecular formula?
Solution
Determining the molecular formula from the provided data will require comparison of the compound’s empirical formula mass to its molar mass. As the first step, use the percent composition to derive the compound’s empirical formula. Assuming a convenient, a 100-g sample of nicotine yields the following molar amounts of its elements:
\[\begin{alignat}{2}
&\mathrm{(74.02\:g\: C)\left(\dfrac{1\:mol\: C}{12.01\:g\: C}\right)}&&= \:\mathrm{6.163\:mol\: C}\\
&\mathrm{(8.710\:g\: H)\left(\dfrac{1\:mol\: H}{1.01\:g\: H}\right)}&&= \:\mathrm{8.624\:mol\: H}\\
&\mathrm{(17.27\:g\: N)\left(\dfrac{1\:mol\: N}{14.01\:g\: N}\right)}&&= \:\mathrm{1.233\:mol\: N}
\end{alignat} \nonumber \]
Next, we calculate the molar ratios of these elements relative to the least abundant element, \(\ce{N}\).
\(6.163 \: \text{mol} \: \ce{C}\) / \(1.233 \: \text{mol} \: \ce{N}\) = 5
\(8.264 \: \text{mol} \: \ce{H}\) / \(1.233 \: \text{mol} \: \ce{N}\) = 7
\(1.233 \: \text{mol} \: \ce{N}\) / \(1.233\: \text{mol} \: \ce{N}\) = 1
1.233/1.233 = \(1.000 \: \text{mol} \: \ce{N}\)
6.163/1.233 = \(4.998 \: \text{mol} \: \ce{C}\)
8.624/1.233 = \(6.994 \: \text{mol} \: \ce{H}\)
The C-to-N and H-to-N molar ratios are adequately close to whole numbers, and so the empirical formula is C 5 H 7 N. The empirical formula mass for this compound is therefore 81.13 amu/formula unit, or 81.13 g/mol formula unit.
We calculate the molar mass for nicotine from the given mass and molar amount of compound:
\[\mathrm{\dfrac{40.57\:g\: nicotine}{0.2500\:mol\: nicotine}=\dfrac{162.3\:g}{mol}} \nonumber \]
Comparing the molar mass and empirical formula mass indicates that each nicotine molecule contains two formula units:
\[\mathrm{\dfrac{162.3\:g/mol}{81.13\:\dfrac{g}{formula\: unit}}=2\:formula\: units/molecule} \nonumber \]
Thus, we can derive the molecular formula for nicotine from the empirical formula by multiplying each subscript by two:
\[\ce{(C5H7N)2}=\ce{C10H14N2} \nonumber \]
What is the molecular formula of a compound with a percent composition of 49.47% C, 5.201% H, 28.84% N, and 16.48% O, and a molecular mass of 194.2 amu?
- Answer
-
C 8 H 10 N 4 O 2
Summary
The chemical identity of a substance is defined by the types and relative numbers of atoms composing its fundamental entities (molecules in the case of covalent compounds, ions in the case of ionic compounds). A compound’s percent composition provides the mass percentage of each element in the compound, and it is often experimentally determined and used to derive the compound’s empirical formula. The empirical formula mass of a covalent compound may be compared to the compound’s molecular or molar mass to derive a molecular formula.
Key Equations
- \(\mathrm{\%X=\dfrac{mass\: X}{mass\: compound}\times100\%}\)
- \(\mathrm{\dfrac{molecular\: or\: molar\: mass\left(amu\: or\:\dfrac{g}{mol}\right)}{empirical\: formula\: mass\left(amu\: or\:\dfrac{g}{mol}\right)}=\mathit n\: formula\: units/molecule}\)
- (A x B y ) n = A nx B ny
Glossary
- percent composition
- percentage by mass of the various elements in a compound
- empirical formula mass
- sum of average atomic masses for all atoms represented in an empirical formula


2.1: Matrix Arithmetic
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Outcomes
- Perform the matrix operations of matrix addition, scalar multiplication, transposition and matrix multiplication. Identify when these operations are not defined. Represent these operations in terms of the entries of a matrix.
- Prove algebraic properties for matrix addition, scalar multiplication, transposition, and matrix multiplication. Apply these properties to manipulate an algebraic expression involving matrices.
- Compute the inverse of a matrix using row operations, and prove identities involving matrix inverses.
- Solve a linear system using matrix algebra.
- Use multiplication by an elementary matrix to apply row operations.
- Write a matrix as a product of elementary matrices.
You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications.
Recall that a matrix is a rectangular array of numbers. Several of them are referred to as matrices . For example, here is a matrix.
\[\left[ \begin{array}{rrrr} 1 & 2 & 3 & 4 \\ 5 & 2 & 8 & 7 \\ 6 & -9 & 1 & 2 \end{array} \right] \label{matrix}\]
Recall that the size or dimension of a matrix is defined as \(m\times n\) where \(m\) is the number of rows and \(n\) is the number of columns. The above matrix is a \(3\times 4\) matrix because there are three rows and four columns. You can remember the columns are like columns in a Greek temple. They stand upright while the rows lay flat like rows made by a tractor in a plowed field.
When specifying the size of a matrix, you always list the number of rows before the number of columns.You might remember that you always list the rows before the columns by using the phrase Row man C atholic.
Consider the following definition.
There is some notation specific to matrices which we now introduce. We denote the columns of a matrix \(A\) by \(A_{j}\) as follows
\[A = \left[ \begin{array}{rrrr} A_{1} & A_{2} & \cdots & A_{n} \end{array} \right]\nonumber \] Therefore, \(A_{j}\) is the \(j^{th}\) column of \(A\) , when counted from left to right.
The individual elements of the matrix are called entries or components of \(A\) . Elements of the matrix are identified according to their position. The \(\mathbf{\left( i, j \right)}\) -entry of a matrix is the entry in the \(i^{th}\) row and \(j^{th}\) column. For example, in the matrix \(\eqref{matrix}\) above, \(8\) is in position \(\left(2,3 \right)\) (and is called the \(\left(2,3 \right)\) -entry) because it is in the second row and the third column.
In order to remember which matrix we are speaking of, we will denote the entry in the \(i^{th}\) row and the \(j^{th}\) column of matrix \(A\) by \(a_{ij}\) . Then, we can write \(A\) in terms of its entries, as \(A= \left[ a_{ij} \right]\) . Using this notation on the matrix in \(\eqref{matrix}\), \(a_{23}=8, a_{32}=-9, a_{12}=2,\) etc.
There are various operations which are done on matrices of appropriate sizes. Matrices can be added to and subtracted from other matrices, multiplied by a scalar, and multiplied by other matrices. We will never divide a matrix by another matrix, but we will see later how matrix inverses play a similar role.
In doing arithmetic with matrices, we often define the action by what happens in terms of the entries (or components) of the matrices. Before looking at these operations in depth, consider a few general definitions.
One possible zero matrix is shown in the following example.
Note there is a \(2\times 3\) zero matrix, a \(3\times 4\) zero matrix, etc. In fact there is a zero matrix for every size!
In other words, two matrices are equal exactly when they are the same size and the corresponding entries are identical. Thus \[\left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right] \neq \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\nonumber \] because they are different sizes. Also, \[\left[ \begin{array}{rr} 0 & 1 \\ 3 & 2 \end{array} \right] \neq \left[ \begin{array}{rr} 1 & 0 \\ 2 & 3 \end{array} \right]\nonumber \] because, although they are the same size, their corresponding entries are not identical.
In the following section, we explore addition of matrices.
Addition of Matrices
When adding matrices, all matrices in the sum need have the same size. For example, \[\left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \\ 5 & 2 \end{array} \right]\nonumber \] and \[\left[ \begin{array}{rrr} -1 & 4 & 8\\ 2 & 8 & 5 \end{array} \right]\nonumber \] cannot be added, as one has size \(3 \times 2\) while the other has size \(2 \times 3\) .
However, the addition \[\left[ \begin{array}{rrr} 4 & 6 & 3\\ 5 & 0 & 4\\ 11 & -2 & 3 \end{array} \right] + \left[ \begin{array}{rrr} 0 & 5 & 0 \\ 4 & -4 & 14 \\ 1 & 2 & 6 \end{array} \right]\nonumber \] is possible.
The formal definition is as follows.
This definition tells us that when adding matrices, we simply add corresponding entries of the matrices. This is demonstrated in the next example.
Example \(\PageIndex{2}\): Addition of Matrices of Same Size
Add the following matrices, if possible. \[A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 0 & 4 \end{array} \right], B = \left[ \begin{array}{rrr} 5 & 2 & 3 \\ -6 & 2 & 1 \end{array} \right]\nonumber \]
Solution
Notice that both \(A\) and \(B\) are of size \(2 \times 3\) . Since \(A\) and \(B\) are of the same size, the addition is possible. Using Definition \(\PageIndex{4}\) , the addition is done as follows. \[A + B = \left[ \begin{array}{rrr} 1 & 2 & 3 \\ 1 & 0 & 4 \end{array} \right] + \left[ \begin{array}{rrr} 5 & 2 & 3 \\ -6 & 2 & 1 \end{array} \right] = \left[ \begin{array}{rrr} 1+5 & 2+2 & 3+3 \\ 1+ -6 & 0+2 & 4+1 \end{array} \right] = \left[ \begin{array}{rrr} 6 & 4 & 6 \\ -5 & 2 & 5 \end{array} \right]\nonumber \]
Addition of matrices obeys very much the same properties as normal addition with numbers. Note that when we write for example \(A+B\) then we assume that both matrices are of equal size so that the operation is indeed possible.
Proposition \(\PageIndex{1}\): Properties of Matrix Addition
Let \(A,B\) and \(C\) be matrices. Then, the following properties hold.
- Commutative Law of Addition \[A+B=B+A \label{mat1}\]
- Associative Law of Addition \[\left( A+B\right) +C=A+\left( B+C\right) \label{mat2}\]
- Existence of an Additive Identity \[\begin{array}{c} \mbox{There exists a zero matrix 0 such that}\\ A+0=A \label{mat3} \end{array}\]
- Existence of an Additive Inverse \[\begin{array}{c} \mbox{There exists a matrix $-A$ such that} \\ A+\left( -A\right) =0 \label{mat4} \end{array}\]
- Proof
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Consider the Commutative Law of Addition given in \(\eqref{mat1}\). Let \(A,B,C,\) and \(D\) be matrices such that \(A+B=C\) and \(B+A=D.\) We want to show that \(D=C\) . To do so, we will use the definition of matrix addition given in Definition \(\PageIndex{4}\) . Now, \[c_{ij}=a_{ij}+b_{ij}=b_{ij}+a_{ij}=d_{ij}\nonumber \] Therefore, \(C=D\) because the \(ij^{th}\) entries are the same for all \(i\) and \(j\) . Note that the conclusion follows from the commutative law of addition of numbers, which says that if \(a\) and \(b\) are two numbers, then \(a+b = b+a\) . The proof of the other results are similar, and are left as an exercise.
We call the zero matrix in \(\eqref{mat3}\) the additive identity . Similarly, we call the matrix \(-A\) in \(\eqref{mat4}\) the additive inverse . \(-A\) is defined to equal \(\left( -1\right) A = [-a_{ij}].\) In other words, every entry of \(A\) is multiplied by \(-1\) .
In the next section we will study scalar multiplication in more depth to understand what is meant by \(\left( -1\right) A.\)
Scalar Multiplication of Matrices
Recall that we use the word scalar when referring to numbers. Therefore, scalar multiplication of a matrix is the multiplication of a matrix by a number. To illustrate this concept, consider the following example in which a matrix is multiplied by the scalar \(3\) . \[3\left[ \begin{array}{rrrr} 1 & 2 & 3 & 4 \\ 5 & 2 & 8 & 7 \\ 6 & -9 & 1 & 2 \end{array} \right] = \left[ \begin{array}{rrrr} 3 & 6 & 9 & 12 \\ 15 & 6 & 24 & 21 \\ 18 & -27 & 3 & 6 \end{array} \right]\nonumber \]
The new matrix is obtained by multiplying every entry of the original matrix by the given scalar.
The formal definition of scalar multiplication is as follows.
Consider the following example.
Example \(\PageIndex{3}\): Effect of Multiplication by a Scalar
Find the result of multiplying the following matrix \(A\) by \(7\) . \[A=\left[ \begin{array}{rr} 2 & 0 \\ 1 & -4 \end{array} \right]\nonumber \]
Solution
By Definition \(\PageIndex{5}\) , we multiply each element of \(A\) by \(7\) . Therefore, \[7A = 7\left[ \begin{array}{rr} 2 & 0 \\ 1 & -4 \end{array} \right] = \left[ \begin{array}{rr} 7(2) & 7(0) \\ 7(1) & 7(-4) \end{array} \right] = \left[ \begin{array}{rr} 14 & 0 \\ 7 & -28 \end{array} \right]\nonumber \]
Similarly to addition of matrices, there are several properties of scalar multiplication which hold.
Proposition \(\PageIndex{2}\): Properties of Scalar Multiplication
Let \(A, B\) be matrices, and \(k, p\) be scalars. Then, the following properties hold.
- Distributive Law over Matrix Addition \[k \left( A+B\right) =k A+ kB\nonumber \]
- Distributive Law over Scalar Addition \[\left( k +p \right) A= k A+p A\nonumber \]
- Associative Law for Scalar Multiplication \[k \left( p A\right) = \left( k p \right) A\nonumber \]
- Rule for Multiplication by \(1\) \[1A=A\nonumber \]
- Proof
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The proof of this proposition is similar to the proof of Proposition \(\PageIndex{1}\) and is left an exercise to the reader.


11.15: Stellar and Celestial Object Brightness
There are a number of other stellar characteristics astronomers and astrophysicists consider. Luminosity is a star’s total amount of power radiated into space, measured in watts. Think about a light bulb in your home. The output is measured in watts. Most of us use 60 watts tungsten light bulbs or the new LED bulbs with an output of around 60 watts, yet using about 10 watts of power.
How does a light bulb with a luminosity of 60 watts compare to the Sun’s output? The accepted luminosity of the Sun is 3. 846×10 26 watts. With such large numbers, the Sun’s luminosity is often stated as 1 L ⊙ . The number 1 L ⊙ is easier to work with than 3. 846×10 26 watts, yet they mean the same quantity.
Apparent Brightness is the amount of light reaching us per unit area, measured in flux. This incoming light varies by the Inverse Square Law . Mathematically this is given as:
Apparent Brightness is proportional to 1/d 2
Let’s look at an example many have experienced. You see a light one mile away, and note its brightness. You move away from the light, and you again look at the light. You might think the light should be half as bright. Actually it is ¼ as bright, as given by the Inverse Square Law.
Apparent brightness at 1 mile: 1/1 2 = 1 Apparent brightness at 2 miles: 1/2 2 = ¼
We all experience the Inverse Square Law in real life, from driving a car at night to looking at lights on and in buildings and streetlights as we approach a light or move away from the light.


4.18: Assignment- Create Document with Tables and Images
For this assignment, you will format a document with a table and an image. Follow the directions, then submit your assignment. If you get stuck on a step, review this module or previous modules and ask your classmates for help in the discussion forum.
- Download the report document and open it in Microsoft Word.
- Save the document as BA132_LastName_Report.docx , replacing “LastName” with your own last name. (Example: BA132_Hywater_Newsletter). It is a good idea to save your work periodically.
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Convert the text after “The top ten are listed in the table below” and before “The Retail Industry” heading into a table. Separate text at Tabs.
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Apply any table style.
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Download this image to your computer.
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Insert the shopping cart image into the document at the beginning of the first paragraph (It starts with “Retailing is important to understand”).
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Apply a square text wrap and move the image so it is aligned with the right side of the document.
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Optional: Resize the image. (Remember, hold down the Shift key while dragging a corner to keep the image proportional.)
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Download this graph image to your computer.
- Insert the graph image into your document after “creating greater contact efficiency for consumers.”
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Add a top and bottom text wrap to the graph image.
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Look for the phrase “You can view the number of jobs and retail presence in your state at the National Retail Federation (NRF)” in your document. Add a hyperlink to “National Retail Federation (NRF)” that leads to this website:
https://nrf.com/advocacy/retails-impact
- Save your document again and submit your work in your course online.
Contributors and Attributions
CC licensed content, Shared previously
- shopping-business-retail [used in report document]. Authored by : Alexas_Fotos. Located at : pixabay.com/en/shopping-business-retail-1165437/. License : CC0: No Rights Reserved
- Reading: Types of Retailers. Authored by : Lumen Learning. Located at : https://courses.lumenlearning.com/marketing-spring2016/chapter/reading-types-of-retailers/ . License : CC BY: Attribution
Lumen Learning authored content
- Assignment: Report. Authored by : Lumen Learning. License : CC BY: Attribution
- The Retail Industry [used in report document]. Authored by : Lumen Learning. Located at : https://courses.lumenlearning.com/marketing-spring2016/chapter/reading-define-retailing/ . License : CC BY: Attribution
- Outcome: Retailers As Channels of Distribution. Authored by : Lumen Learning. Located at : https://courses.lumenlearning.com/marketing-spring2016/chapter/outcome-retailers-as-channels-of-distribution/ . License : CC BY: Attribution


1.3: A Picture of Sara Baartman
Another problematic arena in the field of human sexuality has been well documented examples of ethical violations that stem from both individual and structural racism, dating back to the development of the field up until present day. We begin with the case of enslaved woman, Sara Baartman. Today, she is seen by many as the epitome of colonial exploitation and racism, of the ridicule and commodification of Black bodies. Sarah Baartman was brought to Europe seemingly on false pretenses by a British doctor, stage-named the "Hottentot Venus." She was paraded around "freak shows" in London and Paris, with crowds invited to look at her large buttocks. Baartman died on 29 December 1815, but her exhibition continued. Her brain, skeleton and sexual organs remained on display in a Paris museum until 1974. Her remains weren't repatriated and buried until 2002 (Parkinson, 2016). A 2018 exhibition featuring a sculpture of Sarah Baartman at the University of Cape Town, South Africa was created to facilitate dialogue about heritage and contemporary life.
A second example is Dr. J. Marion Sims, who has been heralded as the father of modern gynecology. It has long been asserted that Dr. Sims perfected the first consistently successful operation for the cure of “vesicovaginal fistula (VVF), a catastrophic complication of childbirth in which a hole develops between a woman's bladder and her vagina and leads to constant, unremitting, and uncontrollable urinary incontinence” (Wall, 2006). He did so by practicing on slave women without the use of anesthesia, which clearly raises questions regarding Dr. Sim’s medical ethics and his perceptions surrounding the human rights of enslaved persons. During Sim's tenure, women suffered horribly from this condition and a race for the cure was critical. Dr Sims' use of slave women was well documented historically. By the very nature of their enslavement, these women were unable to consent. He performed surgery on them as test cases, and then performed more surgeries later on some white women without anesthesia. The use of anesthesia was new during this time, and anesthesia did not become commonplace until he was routinely performing this procedure on white women (Wall, 2006). He later described white women complaining of pain while enslaved Black women did not, and used this anecdotal evidence to make assumptions about differing pain thresholds based on race, rather than other possible factors for their differing responses, such as enslaved women’s likely fear of punishment for expressing said pain. Some argue it is not because Dr. Sims was racist, but rather, that anesthesia was not widely used; however, whatever biases Simms had about pain thresholds and racial identity allowed him to make assumptions about how the women he worked on experienced the procedures.
There is much controversy surrounding the practices of Dr. Sims and the ethical considerations factoring in despite the invention’s historical significance. A vesicovaginal fistula (VVF), rendered women greatly incapacitated during the time Dr. Sims was in practice. It was considered a fate worse than death for many. Prior to Dr. Sims’s breakthrough, there had been no consistently effective procedure to repair the fistula, and there was not a field of gynecology at the time; in fact, examining women’s organs was considered repugnant by male doctors. From 1845-1849, Dr. Sims performed experimental surgery on 7 enslaved Black women in his small backyard hospital (Ojanuga, 1993). He performed his experimental surgery on his first patient Lucy in front of 12 male doctors watching as she crouched on her hands and knees and endured the hour long procedure without anesthesia. The surgery was a failure and she almost died. Another patient, an enslaved woman named Anarcha endured 13 operations without anesthesia with limited success (Ojanuga, 1993). Dr. Sims attempted his procedure on white women but as previously mentioned, found that they could not endure the pain. In the minds of many who look at the history of this particular doctor and his practices, Dr. Sims was not merely a product of his time but rather his infliction of pain on Black women was unethical, cruel and callous.
Sadly, the pattern of white male doctors who callously practice unethical procedures on women and people of color does not end with Dr. Simms. Well into the 21st century, forced sterilization of women in California correctional institutions was a consistent practice (Jindia, 2020).
A state audit and prison records reveal nearly 1,400 sterilizations between 1997 and 2013. In addition to people sterilized during labor, an unknown number of cis women and trans people were sterilized during other abdominal procedures (Jindia, 2020).
These sterilizations took place without consent or under coercive circumstances. Despite ethical mandates and procedural efforts to make sure research participants are protected, there remain recent examples of gross violations. As we take a look at these, we often find that marginalized populations are the ones most at risk. There seems to almost be permission to test on incarcerated people as if they are lab rats. From high risk cancer treatments, to testing skin cream and cosmetics, they have been used for non-consensual medical testing as if their human rights are not valid. These forced sterilizations of incarcerated women in California prisons are the topic of the 2021 documentary, Belly Of The Beast | Documentary Film | Official Trailer . This film highlights one of the ways in which the lives of incarcerated people are not their own, and basic human rights to choice and autonomy of their bodies were grossly disregarded. The 13th Amendment to the Constitution allows for involuntary servitude as a punishment for crime as a means to take away any free will incarcerated people have thus this is just another link in the chain of enslavement. But it is not just the incarcerated who have their free will disregarded. Those institutionalized in mental health facilities as well have been sterilized against their will. Many of these facilities were shut down when then President Ronald Reagan gutted much of the mental health services and facilities. While these facilities were not without problems, this forced many who needed support out on their own, often ending up in prison as they were incapable of getting what they needed to live independently. Facilities for the mentally ill in many ways were like prisons in that those housed there were not treated humanely. A recent expose highlighted the decades long practice of forced sterilization and castration of men and women who were living in one of these facilities, the Sonoma home, one of the many homes for the mentally ill/disabled that practiced this type of forced medical intervention (Barber, 2021). The medical directors chose to undertake these sterilizations based solely on their prejudiced attitudes towards the people whose care they were charged with. The unfortunate truth is that white supremacy, sexism and ableism were deeply entrenched in the mindsets of the medical community that made these decisions without consent, and this influences the science and decision making practices of the medical community to this day.
Just as the enslaved women could not consent, incarcerated and institutionalized people face similar lack of rights, and are coerced via the threat of increased sentences if they fail to comply with authority. The legacy of structural racism, sexism and heteronormativity are part and parcel of the field of Human Sexuality. There is much more to explore about the ways in which what is otherwise a rich field has to reckon with its problematic past and present. Its value stretches far beyond the walls of the academy and has real world consequences that affect all of humanity. A deeper understanding of human sexuality will help us put all of this in context so let's now turn our attention to understanding the field at large.


2.23: Chorale
Before we can dive into a study of Bach’s cantatas, we need to learn about an important element of Lutheran sacred music: the chorale. Bach worked these chorales or hymn tunes into many of his church cantatas, and the three movements from Cantata 140 Wachet auf all make use of the chorale tune on which that cantata is based.
Introduction
A chorale is a melody to which a hymn is sung by a congregation in a German Protestant Church service. The typical four-part setting of a chorale, in which the sopranos (and the congregation) sing the melody along with three lower voices, is known as a chorale harmonization . In certain modern usage, this term may include classical settings of such hymns and works of a similar character.
Chorales tend to be simple and singable tunes. The words are often sung to a rhyming scheme and are in a strophic form (the same melody used for different verses). Within a verse, many chorales follow the AAB pattern of melody that is known as the German bar form.
History
Starting in 1523, Martin Luther began translating worship texts into German from the Latin, so that the people could understand, continue to learn and participate. This created an immediate need for a large repertoire of new chorales. He composed some chorales himself, such as A Mighty Fortress . For other chorales he used Gregorian chant melodies used in Roman Catholic worship and fitted them with new German texts, sometimes adapting the same melody more than once. For example, he fitted the melody of the hymn “Veni redemptor gentium” to three different texts, “Verleih uns Frieden gnädiglich,” “Erhalt uns, Herr, bei deinem Wort,” and “Nun komm, der Heiden Heiland.” A famous example is “Christ lag in Todes Banden,” which is based on the tune of the Catholic Easter Sequence, “Victimae Paschali Laudes.” As early as 1524, Johann Walter published Eyn geystlich Gesangk Buchleyn , the first hymnal for choir, in Wittenberg.
Johann Sebastian Bach harmonised hundreds of chorales, typically used at the end of his cantatas and concluding scenes in his Passions. In his St Matthew Passion , he set five stanzas of “O Haupt voll Blut und Wunden” in four different ways. He also used hymns as the base for his cycle ofchorale cantatas and chorale preludes. Bach concentrated on the chorales especially in the Chorale cantatas of his second annual cycle, composed mostly in 1724–25.
Today, many of the Lutheran chorales are familiar as hymns used in Protestant churches, sometimes sung in four-voice harmony.
Derived Forms
Chorales also appear in chorale preludes, pieces generally for organ designed to be played immediately before the congregational singing of the hymn. A chorale prelude includes the melody of the chorale, and adds contrapuntal lines. One of the first composers to write chorale preludes was Samuel Scheidt. Bach’s many chorale preludes are the best-known examples of the form. Later composers of the chorale prelude include Johannes Brahms, such as Eleven Chorale Preludes, and Max Reger who composed Wie schön leucht’ uns der Morgenstern on Nicolai’s hymn, among many others.
Anton Bruckner make frequent use of the chorale as a compositional device, based on his understanding of musical settings of the liturgy and Johann Sebastian Bach’s chorale preludes. He used it in his symphonies, masses and motets, for example Dir, Herr, dir will ich mich ergeben and In jener letzten der Nächte , often in contrast to and combination with the fugue, as in Psalm 22 and in the Finale of Symphony No. 5.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Chorale. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Chorale . License : Public Domain: No Known Copyright


26.1: The Rise of Franklin Roosevelt
Franklin Roosevelt was part of the political establishment and the wealthy elite, but in the 1932 presidential campaign, he did not want to be perceived that way. Roosevelt felt that the country needed sweeping change, and he ran a campaign intended to convince the American people that he could deliver that change. It was not the specifics of his campaign promises that were different; in fact, he gave very few details and likely did not yet have a clear idea of how he would raise the country out of the Great Depression. But he campaigned tirelessly, talking to thousands of people, appearing at his party’s national convention, and striving to show the public that he was a different breed of politician. As Hoover grew more morose and physically unwell in the face of the campaign, Roosevelt thrived. He was elected in a landslide by a country ready for the change he had promised.
THE ELECTION OF FRANKLIN ROOSEVELT
By the 1932 presidential election, Hoover’s popularity was at an all-time low. Despite his efforts to address the hardships that many Americans faced, his ineffectual response to the Great Depression left Americans angry and ready for change. Franklin Roosevelt, though born to wealth and educated at the best schools, offered the change people sought. His experience in politics had previously included a seat in the New York State legislature, a vice-presidential nomination, and a stint as governor of New York. During the latter, he introduced many state-level reforms that later formed the basis of his New Deal as well as worked with several advisors who later formed the Brains Trust that advised his federal agenda.
Roosevelt exuded confidence, which the American public desperately wished to see in their leader (Figure 26.1.2). And, despite his affluence, Americans felt that he could relate to their suffering due to his own physical hardships; he had been struck with polio a decade earlier and was essentially paralyzed from the waist down for the remainder of his life. Roosevelt understood that the public sympathized with his ailment; he likewise developed a genuine empathy for public suffering as a result of his illness. However, he never wanted to be photographed in his wheelchair or appear infirm in any way, for fear that the public’s sympathy would transform into concern over his physical ability to discharge the duties of the Oval Office.
Roosevelt also recognized the need to convey to the voting public that he was not simply another member of the political aristocracy. At a time when the country not only faced its most severe economic challenges to date, but Americans began to question some of the fundamental principles of capitalism and democracy, Roosevelt sought to show that he was different—that he could defy expectations—and through his actions could find creative solutions to address the nation’s problems while restoring public confidence in fundamental American values. As a result, he not only was the first presidential candidate to appear in person at a national political convention to accept his party’s nomination but also flew there through terrible weather from New York to Chicago in order to do so—a risky venture in what was still the early stages of flight as public transportation. At the Democratic National Convention in 1932, he coined the famous phrase: “I pledge myself to a new deal for the American people.” The New Deal did not yet exist, but to the American people, any positive and optimistic response to the Great Depression was a welcome one.
Hoover assumed at first that Roosevelt would be easy to defeat, confident that he could never carry the eastern states and the business vote. He was sorely mistaken. Everywhere he went, Hoover was met with antagonism; anti-Hoover signs and protests were the norm. Hoover’s public persona declined rapidly. Many news accounts reported that he seemed physically unwell, with an ashen face and shaking hands. Often, he seemed as though he would faint, and an aide constantly remained nearby with a chair in case he fell. In contrast, Roosevelt thrived on the campaign. He commented, “I have looked into the faces of thousands of Americans, and they have the frightened look of lost children.”
The election results that November were never really in question: With three million more people voting than in 1928, Roosevelt won by a popular count of twenty-three million to fifteen million. He carried all but six states while winning over 57 percent of the popular vote. Whether they voted due to animosity towards Hoover for his relative inactivity, or out of hope for what Roosevelt would accomplish, the American public committed themselves to a new vision. Historians identify this election as the beginning of a new Democratic coalition, bringing together African Americans, other ethnic minorities, and organized labor as a voting bloc upon whom the party would rely for many of its electoral victories over the next fifty years. Unlike some European nations where similar challenges caused democratic constitutions to crumble and give way to radical ideologies and authoritarian governments, the Roosevelt administration changed the nation’s economic fortunes with reforms, preserved the constitution, and expanded rather than limited the reach of democratic principles into the market economy. As a result, radical alternatives, such as the Fascist movement or Communist Party, remained on the margins of the nation’s political culture.
THE INTERREGNUM
After the landslide election, the country—and Hoover—had to endure theinterregnum, the difficult four months between the election and President Roosevelt’s inauguration in March 1933.  Roosevelt remained gracious but refused to begin his administration as the incumbent’s advisor without any legal authority necessary to change policy. Unwilling to tie himself to Hoover’s legacy of failed policies, Roosevelt kept quiet when Hoover supported the passage of a national sales tax. Meanwhile, the country suffered from Hoover’s inability to further drive a legislative agenda through Congress. It was the worst winter since the beginning of the Great Depression, and the banking sector once again suffered another round of panics. While Roosevelt kept his distance from the final tremors of the Hoover administration, the country continued to suffer in wait. In part as a response to the challenges of this time, the U.S. Constitution was subsequently amended to reduce the period from election to inauguration to the now-commonplace two months.
Any ideas that Roosevelt held almost did not come to fruition, thanks to a would-be assassin’s bullet. On February 15, 1933, after delivering a speech from his open car in Miami’s Bayfront Park, local Italian bricklayer Giuseppe Zangara emerged from a crowd of well-wishers to fire six shots from his revolver. Although Roosevelt emerged from the assassination attempt unscathed, Zangara wounded five individuals that day, including Chicago Mayor Tony Cermak, who attended the speech in the hopes of resolving any long-standing differences with the president-elect. Roosevelt and his driver immediately rushed Cermak to the hospital where he died three days later. Roosevelt’s calm and collected response to the event reassured many Americans of his ability to lead the nation through the challenges they faced. All that awaited was Roosevelt’s inauguration before his ideas would unfold to the expectant public.
So what was Roosevelt’s plan? Before he took office, it seems likely that he was not entirely sure. Certain elements were known: He believed in positive government action to solve the Depression; he believed in federal relief, public works, social security, and unemployment insurance; he wanted to restore public confidence in banks; he wanted stronger government regulation of the economy; and he wanted to directly help farmers. But how to take action on these beliefs was more in question. A month before his inauguration, he said to his advisors, “Let’s concentrate upon one thing: Save the people and the nation, and if we have to change our minds twice every day to accomplish that end, we should do it.”
Unlike Hoover, who professed an ideology of “American individualism,” an adherence that rendered him largely incapable of widespread action, Roosevelt remained pragmatic and open-minded to possible solutions. To assist in formulating a variety of relief and recovery programs, Roosevelt turned to a group of men who had previously orchestrated his election campaign and victory. Collectively known as the “Brains Trust” (a phrase coined by a New York Times reporter to describe the multiple “brains” on Roosevelt’s advisory team), the group most notably included Rexford Tugwell, Raymond Moley, and Adolph Berle. Moley, credited with bringing the group into existence, was a government professor who advocated for a new national tax policy to help the nation recover from its economic woes. Tugwell, who eventually focused his energy on the country’s agricultural problems, saw an increased role for the federal government in setting wages and prices across the economy. Berle was a mediating influence, who often advised against a centrally controlled economy, but did see the role that the federal government could play in mediating the stark cycles of prosperity and depression that, if left unchecked, could result in the very situation in which the country presently found itself. Together, these men, along with others, advised Roosevelt through the earliest days of the New Deal and helped to craft significant legislative programs for congressional review and approval.
INAUGURATION DAY: A NEW BEGINNING
March 4, 1933, dawned gray and rainy. Roosevelt rode in an open car along with outgoing president Hoover, facing the public, as he made his way to the U.S. Capitol. Hoover’s mood was somber, still personally angry over his defeat in the general election the previous November; he refused to crack a smile at all during the ride among the crowd, despite Roosevelt’s urging to the contrary. At the ceremony, Roosevelt rose with the aid of leg braces equipped under his specially tailored trousers and placed his hand on a Dutch family Bible as he took his solemn oath. At that very moment, the rain stopped and the sun began to shine directly on the platform, and those present would later claim that it was as though God himself was shining down on Roosevelt and the American people in that moment (Figure 26.1.3).
Bathed in the sunlight, Roosevelt delivered one of the most famous and oft-quoted inaugural addresses in history. He encouraged Americans to work with him to find solutions to the nation’s problems and not to be paralyzed by fear into inaction. Borrowing a wartime analogy provided by Moley, who served as his speechwriter at the time, Roosevelt called upon all Americans to assemble and fight an essential battle against the forces of economic depression. He famously stated, “The only thing we have to fear is fear itself.” Upon hearing his inaugural address, one observer in the crowd later commented, “Any man who can talk like that in times like these is worth every ounce of support a true American has.” To borrow the popular song title of the day, “happy days were here again.” Foregoing the traditional inaugural parties, the new president immediately returned to the White House to begin his work to save the nation.
Visit the American Presidency Project to listen to Roosevelt’s first inaugural speech and identify ways he conveyed optimism and a spirit of community to his listeners.
Section Summary
Franklin Roosevelt was a wealthy, well-educated, and popular politician whose history of polio made him a more sympathetic figure to the public. He did not share any specifics of his plan to bring the country out of the Great Depression, but his attitude of optimism and possibility contrasted strongly with Hoover’s defeated misery. The 1932 election was never really in question, and Roosevelt won in a landslide. During the four-month interregnum, however, Americans continued to endure President Hoover’s failed policies, which led the winter of 1932–1933 to be the worst of the Depression, with unemployment rising to record levels.
When Roosevelt took office in March 1933, he infused the country with a sense of optimism. He still did not have a formal plan but rather invited the American people to join him in the spirit of experimentation. Roosevelt did bring certain beliefs to office: the belief in an active government that would take direct action on federal relief, public works, social services, and direct aid to farmers. But as much as his policies, Roosevelt’s own personality and engaging manner helped the country feel that they were going to get back on track.
Review Questions
Which of the following best describes Roosevelt’s attempts to push his political agenda in the last months of Hoover’s presidency?
Roosevelt spoke publicly on the issue of direct relief.
Roosevelt met privately with Hoover to convince him to institute certain policy shifts before his presidency ended.
Roosevelt awaited his inauguration before introducing any plans.
Roosevelt met secretly with members of Congress to attempt to win their favor.
C
Which of the following policies did Roosevelt not include among his early ideas for a New Deal?
public works
government regulation of the economy
elimination of the gold standard
aid to farmers
D
What was the purpose of Roosevelt’s “Brains Trust?”
Roosevelt recruited his “Brains Trust” to advise him in his inception of a variety of relief and recovery programs. Among other things, the members of this group pushed for a new national tax policy; addressed the nation’s agricultural problems; advocated an increased role for the federal government in setting wages and prices; and believed that the federal government could temper the boom-and-bust cycles that rendered the economy unstable. These advisors helped to craft the legislative programs that Roosevelt presented to Congress.
Glossary
- Brains Trust
- an unofficial advisory cabinet to President Franklin Roosevelt, originally gathered while he was governor of New York, to present possible solutions to the nations’ problems; among its prominent members were Rexford Tugwell, Raymond Moley, and Adolph Berle
- interregnum
- the period between the election and the inauguration of a new president; when economic conditions worsened significantly during the four-month lag between Roosevelt’s win and his move into the Oval Office, Congress amended the Constitution to limit this period to two months


32.1: Introduction
Ecology
Ecology is the study of the interactions of living organisms with their environment. One core goal of ecology is to understand the distribution and abundance of living things in the physical environment. Attainment of this goal requires the integration of scientific disciplines inside and outside of biology, such as mathematics, statistics, biochemistry, molecular biology, physiology, evolution, biodiversity, geology, and climatology.
Levels of Ecological Study
When a discipline such as biology is studied, it is often helpful to subdivide it into smaller, related areas. For instance, cell biologists interested in cell signaling need to understand the chemistry of the signal molecules (which are usually proteins) as well as the result of cell signaling. Ecologists interested in the factors that influence the survival of an endangered species might use mathematical models to predict how current conservation efforts affect endangered organisms.
To produce a sound set of management options, a conservation biologist needs to collect accurate data, including current population size, factors affecting reproduction (like physiology and behavior), habitat requirements (such as plants and soils), and potential human influences on the endangered population and its habitat (which might be derived through studies in sociology and urban ecology). Within the discipline of ecology, researchers work at four general levels, which sometimes overlap. These levels are organism, population, community, and ecosystem (Figure 44.2).
Ecosystem & Eutrophication
An ecosystem is defined as an association of life and the physical environment. Ecosystems take into account both living organisms and the nonliving components like water, soil, light, etc. Ecosystems can be either terrestrial (dessert, forest, grassland) or aquatic (coral reef, pond, estuary).
Within ecosystems scientists study energy transformations . Energy typically enters the ecosystem from the sun, is transferred to photosynthetic organisms (primary producers) and then to organisms that need to eat others for energy, the consumers or heterotrophs. An ecosystem will typically have several levels of consumers. The transfer of energy from organism to organism can be illustrated through trophic levels. A trophic level includes all organisms the same number of transfer steps away from the energy input into an ecosystem. A simple trophic level diagram for a forest ecosystem is illustrated below. Notice the photosynthetic organisms like plants are on the first trophic level, one step away from the energy source. The squirrel is a primary consumer (eating plants directly) but on the second tropic level two steps away from the primary energy source. At each step along the pathway some energy is lost as heat. Therefore at higher tropic levels there are fewer organisms (as noted by the pyramid shape) and most ecosystems support no more than four or five total trophic levels.
Ecosystems are changing due to human behaviors. Humans negatively impact natural ecosystems through activities such as deforestation, hunting, and pollution. Today’s lab focuses on the impact of eutrophication on aquatic ecosystems. Through eutrophication bodies of water acquire extremely high concentrations of nutrients. The source of these nutrients can be natural or artificial. Humans cause cultural eutrophication through behaviors like run off from agricultural fields, wastewater from sewage treatment plants, and excel detergents running into bodies of water. The excess nutrients fuel photosynthesis causing an increased growth in algae, a photosynthetic primary producer protist, and an algae bloom. During the bloom the algae cover the surface of the water. When the algae die, decomposers in the ecosystem break down the protists using up the oxygen available in the aquatic environment through respiration. As oxygen levels decrease (hypoxia), fish die, and the balance of the ecosystem is destroyed.
In an effort to understand ecosystems and human impact on ecosystems scientists often use models. Modeling of ecosystems serves two main functions. First, the model represents scientist’s best understanding of the relationships/functions that define the ecosystem. Second, models allow scientists to investigate questions that would be impossible to pursue in reality.
Ecosystem models fall into two categories, analytical models and simulation models. Analytical models use math to explain simple linear relationships. Simulation models, more widely used, are used to illustrate complex non linear relationships in ecosystems. Since the natural ecosystem has numerous interactions between living and non living components ecosystem models must simplify this real world situation. Models incorporate only the most important components or group similar components in an effort to effectively represent the ecosystem in a straightforward fashion.


2.3: Queen Elizabeth
(1533-1603)
Queen Elizabeth I was the daughter of Henry VIII and Anne Boleyn, who was beheaded when Elizabeth was two years old. After Henry and his third wife Jane Seymour (c. 1508-1537) had Edward, Henry’s longedfor male heir, Elizabeth and her half sister Mary were declared illegitimate. As Henry’s daughter, legitimate or otherwise, Elizabeth was tutored in music and in the French, Spanish, Greek, and Latin languages. Her tutors included the classicists William Grindal (d. 1548) and Roger Ascham (c. 1515-1568).
Although she lost her biological mother at an early age, Elizabeth received care from her step-mothers, particularly Catherine Parr (1512-1548), who survived Henry. After Henry’s death, Catherine married Thomas Seymour, who used his proximity to Elizabeth to influence her and possibly to angle for her hand in a future marriage.  The reign of his successor, Elizabeth’s halfsister Mary, however, caused dangerous upheaval, to Elizabeth proper and to the country as a whole. Protestant rebels rallied against the Roman Catholic Mary’s succession, first for Jane Grey (1537-1554), a legitimate daughter of Henry’s sister, then for Elizabeth. Although she herself did not encourage or conspire with the rebels, Elizabeth was briefly imprisoned in the Tower of London under suspicion of treason.
Elizabeth succeeded Mary to rule a country straitened by Continental war and internal religious strife. With the Act of Supremacy (1558), Elizabeth reestablished the Church of England; with the Act of Uniformity (1558), she established the Book of Common Prayer. She also maintained a tenuous peace with the Continent, particularly France and Spain, for the greater part of her reign. Prudent, cautious, and extremely wise in her choice of advisors, Elizabeth used her single status to her advantage. Nations that might have warred with her instead sent her suitors, including the Dukes of Saxony, Holstein, Anjou and Savoy; Prince Frederick of Denmark; Don Carlos of Spain; and King Charles IX of France. Only when she supported the Protestant rebellion against Spain in the Netherlands did she trigger the long-threatened attack of the Spanish Armada in 1588. Its defeat heralded a golden age for her realm and her rule.
Religious strife again threatened when her cousin Mary Stuart, Queen of the Scots, married the future King of France. His early death brought the threats closer to home with several assassination attempts made against Elizabeth in hopes of making Mary Queen of England as well as Scotland. When Mary took refuge in England, Elizabeth had her imprisoned in Fotheringhay Castle, keeping the threat at bay for nineteen years. From her imprisonment, Mary encouraged rebellion and was implicated in the Babington Plot against Elizabeth’s life and was consequently beheaded.
In her later years, Elizabeth faced riots at home due to food shortages and rebellion in Ireland. The latter gave armed troops to a court favorite, Robert Devereaux, 2nd Earl of Essex (b. 1565). His troops did not suppress the Irish rebellion but instead supported his own against Elizabeth. He was beheaded for treason in 1601. Elizabeth I’s long reign fostered England’s economic growth, military strength, and, most importantly, artistic and literary immortality. To the end of her life, she refused to soothe her nation’s fears against her own mortality by marrying and then bearing an heir. She wedded herself to England and her subjects.
Her writing reflects the emotional weight of the political challenges she faced as a female and as a sovereign—as a woman, who was both an individual and a symbol, who always had too much to lose. She wrote in many forms, including speeches for state occasions, translations, and poems. In her poems, we hear her voice—poignant and strong, emotional and moral—and can appreciate her complexity.


3.11: Observations outside Earth's Atmosphere
Learning Objectives
By the end of the section, you will be able to:
- List the advantages of making astronomical observations from space
- Explain the importance of the Hubble Space Telescope
- Describe some of the major space-based observatories astronomers use
Earth’s atmosphere blocks most radiation at wavelengths shorter than visible light, so we can only make direct ultraviolet, X-ray, and gamma ray observations from space (though indirect gamma ray observations can be made from Earth). Getting above the distorting effects of the atmosphere is also an advantage at visible and infrared wavelengths. The stars don’t “twinkle” in space, so the amount of detail you can observe is limited only by the size of your instrument. On the other hand, it is expensive to place telescopes into space, and repairs can present a major challenge. This is why astronomers continue to build telescopes for use on the ground as well as for launching into space.
Airborne and Space Infrared Telescopes
Water vapor, the main source of atmospheric interference for making infrared observations, is concentrated in the lower part of Earth’s atmosphere. For this reason, a gain of even a few hundred meters in elevation can make an important difference in the quality of an infared observatory site. Given the limitations of high mountains, most of which attract clouds and violent storms, and the fact that the ability of humans to perform complex tasks degrades at high altitudes, it was natural for astronomers to investigate the possibility of observing infrared waves from airplanes and ultimately from space.
Infrared observations from airplanes have been made since the 1960s, starting with a 15-centimeter telescope on board a Learjet. From 1974 through 1995, NASA operated a 0.9-meter airborne telescope flying regularly out of the Ames Research Center south of San Francisco. Observing from an altitude of 12 kilometers, the telescope was above 99% of the atmospheric water vapor. More recently, NASA (in partnership with the German Aerospace Center) has constructed a much larger 2.5-meter telescope, called the Stratospheric Observatory for Infrared Astronomy (SOFIA), which flies in a modified Boeing 747SP (Figure \(\PageIndex{1}\)).
To find out more about SOFIA, watch this video provided by NASA’s Armstrong Flight Research Center.
Getting even higher and making observations from space itself have important advantages for infrared astronomy. First is the elimination of all interference from the atmosphere. Equally important is the opportunity to cool the entire optical system of the instrument in order to nearly eliminate infrared radiation from the telescope itself. If we tried to cool a telescope within the atmosphere, it would quickly become coated with condensing water vapor and other gases, making it useless. Only in the vacuum of space can optical elements be cooled to hundreds of degrees below freezing and still remain operational.
The first orbiting infrared observatory, launched in 1983, was the Infrared Astronomical Satellite (IRAS), built as a joint project by the United States, the Netherlands, and Britain. IRAS was equipped with a 0.6-meter telescope cooled to a temperature of less than 10 K. For the first time, the infrared sky could be seen as if it were night, rather than through a bright foreground of atmospheric and telescope emissions. IRAS carried out a rapid but comprehensive survey of the entire infrared sky over a 10-month period, cataloging about 350,000 sources of infrared radiation. Since then, several other infrared telescopes have operated in space with much better sensitivity and resolution due to improvements in infrared detectors. The most powerful of these infrared telescopes is the 0.85-meter Spitzer Space Telescope, which launched in 2003. A few of its observations are shown in Figure \(\PageIndex{2}\).
Hubble Space Telescope
In April 1990, a great leap forward in astronomy was made with the launch of the Hubble Space Telescope (HST). With an aperture of 2.4 meters, this is the largest telescope put into space so far. (Its aperture was limited by the size of the payload bay in the Space Shuttle that served as its launch vehicle.) It was named for Edwin Hubble, the astronomer who discovered the expansion of the universe in the 1920s (whose work we will discuss in the chapters on Galaxies).
HST is operated jointly by NASA’s Goddard Space Flight Center and the Space Telescope Science Institute in Baltimore. It was the first orbiting observatory designed to be serviced by Shuttle astronauts and, over the years since it was launched, they made several visits to improve or replace its initial instruments and to repair some of the systems that operate the spacecraft (Chapter Figure)—though this repair program has now been discontinued, and no more visits or improvements will be made.
With the Hubble, astronomers have obtained some of the most detailed images of astronomical objects from the solar system outward to the most distant galaxies. Among its many great achievements is the Hubble Ultra-Deep Field, an image of a small region of the sky observed for almost 100 hours. It contains views of about 10,000 galaxies, some of which formed when the universe was just a few percent of its current age (Figure \(\PageIndex{3}\)).
The HST’s mirror was ground and polished to a remarkable degree of accuracy. If we were to scale up its 2.4-meter mirror to the size of the entire continental United States, there would be no hill or valley larger than about 6 centimeters in its smooth surface. Unfortunately, after it was launched, scientists discovered that the primary mirror had a slight error in its shape, equal to roughly 1/50 the width of a human hair. Small as that sounds, it was enough to ensure that much of the light entering the telescope did not come to a clear focus and that all the images were blurry. (In a misplaced effort to save money, a complete test of the optical system had not been carried out before launch, so the error was not discovered until HST was in orbit.)
The solution was to do something very similar to what we do for astronomy students with blurry vision: put corrective optics in front of their eyes. In December 1993, in one of the most exciting and difficult space missions ever flown, astronauts captured the orbiting telescope and brought it back into the shuttle payload bay.  The telescope now works as it was intended to, and further missions to it were able to install even more advanced instruments to take advantage of its capabilities.
High-Energy Observatories
Ultraviolet, X-ray, and direct gamma-ray (high-energy electromagnetic wave) observations can be made only from space. Such observations first became possible in 1946, with V2 rockets captured from Germany after World War II. The US Naval Research Laboratory put instruments on these rockets for a series of pioneering flights, used initially to detect ultraviolet radiation from the Sun. Since then, many other rockets have been launched to make X-ray and ultraviolet observations of the Sun, and later of other celestial objects.
Beginning in the 1960s, a steady stream of high-energy observatories has been launched into orbit to reveal and explore the universe at short wavelengths. Among recent X-ray telescopes is the Chandra X-ray Observatory, which was launched in 1999 (Figure \(\PageIndex{4}\)). It is producing X-ray images with unprecedented resolution and sensitivity. Designing instruments that can collect and focus energetic radiation like X-rays and gamma rays is an enormous technological challenge. The 2002 Nobel Prize in physics was awarded to Riccardo Giacconi, a pioneer in the field of building and launching sophisticated X-ray instruments. In 2008, NASA launched the Fermi Gamma-ray Space Telescope, designed to measure cosmic gamma rays at energies greater than any previous telescope, and thus able to collect radiation from some of the most energetic events in the universe.
One major challenge is to design “mirrors” to reflect such penetrating radiation as X-rays and gamma rays, which normally pass straight through matter. However, although the technical details of design are more complicated, the three basic components of an observing system, as we explained earlier in this chapter, are the same at all wavelengths: a telescope to gather up the radiation, filters or instruments to sort the radiation according to wavelength, and some method of detecting and making a permanent record of the observations. Table \(\PageIndex{1}\) lists some of the most important active space observatories that humanity has launched.
Gamma-ray detections can also be made from Earth’s surface by using the atmosphere as the primary detector. When a gamma ray hits our atmosphere, it accelerates charged particles (mostly electrons) in the atmosphere. Those energetic particles hit other particles in the atmosphere and give off their own radiation. The effect is a cascade of light and energy that can be detected on the ground. The VERITAS array in Arizona and the H.E.S.S. array in Namibia are two such ground-based gamma-ray observatories.
| Observatory | Date Operation Began | Bands of the Spectrum | Notes | Website |
|---|---|---|---|---|
| Hubble Space Telescope(HST) | 1990 | visible, UV, IR | 2.4-m mirror; images and spectra | www.hubblesite.org |
| Chandra X-Ray Observatory | 1999 | X-rays | X-ray images and spectra | www.chandra.si.edu |
| XMM-Newton | 1999 | X-rays | X-ray spectroscopy | http://www.cosmos.esa.int/web/xmm-newton |
| International Gamma-Ray Astrophysics Laboratory(INTEGRAL) | 2002 | X- and gamma-rays | higher resolution gamma-ray images | http://sci.esa.int/integral/ |
| Spitzer Space Telescope | 2003 | IR | 0.85-m telescope | www.spitzer.caltech.edu |
| Fermi Gamma-ray Space Telescope | 2008 | gamma-rays | first high-energy gamma-ray observations | fermi.gsfc.nasa.gov |
| Kepler | 2009 | visible-light | planet finder | kepler.nasa.gov |
| Wide-field Infrared Survey Explorer(WISE) | 2009 | IR | whole-sky map, asteroid searches | www.nasa.gov/mission_pages/WISE/main |
| Gaia | 2013 | visible-light | Precise map of the Milky Way | http://sci.esa.int/gaia/ |
| Transiting Exoplanet Survey Satellite (TESS) | 2018 | visible-light | planet finder | http://tess.mit.edu |
Summary
Infrared observations are made with telescopes aboard aircraft and in space, as well as from ground-based facilities on dry mountain peaks. Ultraviolet, X-ray, and gamma-ray observations must be made from above the atmosphere. Many orbiting observatories have been flown to observe in these bands of the spectrum in the last few decades. The largest-aperture telescope in space is the Hubble Space telescope (HST), the most significant infrared telescope is Spitzer, and Chandra and Fermi are the premier X-ray and gamma-ray observatories, respectively.
Contributors and Attributions
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Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).